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In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax's methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax's data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou's framework for studying shock formation to treat a new solution regime in which wave dispersion is not present.

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In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in 1 + 3 spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state p = c 2 s ρ, 0 < c s < 1/3, the background metric + fluid solutions are globally future-stable under small irrotational perturbations of their initial data. In particular, we prove that the perturbed spacetime solutions, which have the topological structure [0, ∞) × T 3 , are future causally geodesically complete. Our analysis is based on a combination of energy estimates and pointwise decay estimates for quasilinear wave equations featuring dissipative inhomogeneous terms. Our main new contribution is showing that when 0 < c s < 1/3, exponential spacetime expansion is strong enough to suppress the formation of fluid shocks. This contrasts against a well-known result of Christodoulou, who showed that in Minkowski spacetime, the corresponding constant-state irrotational fluid solutions are unstable.

We prove a stable singularity formation result, without symmetry assumptions, for solutions to the Einstein-scalar field and Einstein-stiff fluid systems. Our results apply to small perturbations of the spatially flat Friedmann-Lemaître-Robertson-Walker (FLRW) solution with topology (0, ∞)×T 3 . The FLRW solution models a spatially uniform scalar-field/stiff fluid evolving in a spacetime that expands towards the future and that has a "Big Bang" singularity at {0} × T 3 , where its curvature blows up. We place "initial" data on a Cauchy hypersurface Σ ′ 1 that are close, as measured by a Sobolev norm, to the FLRW data induced on {1} × T 3 . We then study the asymptotic behavior of the perturbed solution in the collapsing direction and prove that its basic qualitative and quantitative features closely resemble those of the FLRW solution.In particular, we construct constant mean curvature-transported spatial coordinates for the perturbed solution covering (t, x) ∈ (0, 1] × T 3 and show that it also has a Big Bang at {0} × T 3 , where its curvature blows up. The blow-up confirms Penrose's Strong Cosmic Censorship hypothesis for the "past-half" of near-FLRW solutions. Furthermore, we show that the Einstein equations are dominated by kinetic (time derivative) terms that induce approximately monotonic behavior near the Big Bang, and consequently, various time-rescaled components of the solution converge to functions of x as t ↓ 0.The most difficult aspect of the proof is showing that the solution exists for (t, x) ∈ (0, 1] × T 3 , and to this end, we derive a hierarchy of energy estimates that are allowed to mildly blow-up as t ↓ 0. To close these estimates, it is essential that we are able to rule out more singular energy blow-up, which is in turn tied to the most important ingredient in our analysis: an L 2 −type energy approximate monotonicity inequality that holds for near-FLRW solutions. In the companion article [73], we used the approximate monotonicity to prove a stability result for solutions to linearized versions of the equations. The present article shows that the linear stability result can be upgraded to control the nonlinear terms.1 Globally hyperbolic spacetimes are those containing a Cauchy hypersurface. 2 The assumptions needed in Hawking's theorem are i) the matter model verifies the strong energy condition, which is T µν − 1 2 (g −1 ) αβ T αβ T µν X µ X ν ≥ 0 whenever X is timelike; ii) k a a < −C everywhere on the initial Cauchy hypersurface, where k is the second fundamental form of Σ t (see (3.6)) and C > 0 is a constant. More precisely, Hawking's theorem shows that no past-directed timelike curve can have length greater than 3 C . In our main results, we derive a slightly sharper version of this estimate that is tailored to the solutions addressed in this article (see inequality (15.6)). 3 As we describe below, the relevant time coordinate t for the perturbed solution has level sets with mean curvature −(1 3)t −1 . 4 Roughly, this conjecture asserts that the maximal globally hyperbolic development...